I am interested in taking the derivative of vector spherical harmonics in the spherical coordinate basis and I was wondering whether anyone had any good resources where to look for help in manipulating things analytically. For example,
$$ \begin{align} \mathbf{A}_{lm}^{(1)}(\hat{\mathbf{r}}) &= \frac{1}{\sqrt{l(l+1)}} \nabla\times(\mathbf{r}Y_{lm}(\hat{\mathbf{r}})) = \frac{1}{\sqrt{l(l+1)}}\nabla Y_{lm}(\hat{\mathbf{r}})\times \mathbf{r} \\ \mathbf{A}_{lm}^{(2)}(\hat{\mathbf{r}}) &= \frac{1}{\sqrt{l(l+1)}}r\nabla Y_{lm}(\hat{\mathbf{r}}) \\ \mathbf{A}_{lm}^{(3)}(\hat{\mathbf{r}}) &= \hat{\mathbf{r}} Y_{lm}(\hat{\mathbf{r}}) \end{align} $$
and I, in particular, am interested in the $\frac{\partial}{\partial\phi}$ derivatives
$$ \begin{align} \frac{\partial}{\partial\phi} \mathbf{A}_{lm}^{(1)}(\hat{\mathbf{r}}) &= ? \\ \frac{\partial}{\partial\phi} \mathbf{A}_{lm}^{(2)}(\hat{\mathbf{r}}) &= ? \\ \frac{\partial}{\partial\phi} \mathbf{A}_{lm}^{(3)}(\hat{\mathbf{r}}) &= im\mathbf{A}_{lm}^{(3)}(\hat{\mathbf{r}}) + \hat{\boldsymbol{\phi}}Y_{lm}(\hat{\mathbf{r}}) + \dots \end{align} $$
Do the terms that give $\hat{\boldsymbol{\phi}}$ from the product-rule when differentiating have a simple analytical expression in terms of the vector spherical harmonics? Any textbook or journal article resources on these derivatives as functions of $r, \theta, \phi?$
Up to some constant factors, I guess the question is whether there the action of the "angular momentum" operator $L_{z}$ on the vector spherical harmonics has a nice analytic form. If so, what are they?
Notations vary. If one looks at Wikipedia, https://en.wikipedia.org/wiki/Vector_spherical_harmonics, then what I am interested in is the azimuthal derivatives of $\mathbf{Y}_{lm}, \boldsymbol{\Psi}_{lm}, \boldsymbol{\Phi}_{lm}$ in the Wikipedia notation.